This lecture concludes our survey of closed holomorphic curves with a discussion, in the first section, of local intersection numbers, positivity of intersections and the adjunction formula for closed holomorphic curves, and then, in the second section, with an explanation of how these figure into the proof of McDuff’s theorem on symplectic ruled surfaces. The last two sections then begin a shift in focus toward punctured holomorphic curves: this discussion starts with a general introduction to contact manifolds and their symplectic fillings and then continues by defining the moduli space of punctured asymptotically cylindrical holomorphic curves in a completed symplectic cobordism between contact manifolds.

This appendix is intended as a quick reference on Siefring’s intersection theory for the benefit of researchers who would like to use it and need an easy place to look up the main facts. All results here are stated without proof, with references (mostly to Lectures 3 and 4 in this book) given for further details. Two additional topics are covered that do not appear elsewhere in this book: covering relations for the star-pairing and normal Chern number, and the intersection product between holomorphic buildings. Finally, the appendix concludes with a comparison of notational and terminology conventions between Siefring’s theory and the equivalent notions that often appear in the literature on embedded contact homology: in particular, we clarify the relationship between Siefring’s relative asymptotic contributions and Hutchings’s asymptotic linking number and writhe.

Using the relative asymptotic results of the previous lecture as a black box, this lecture explains the main definitions and results of Siefring’s intersection theory for punctured holomorphic curves. This includes the definition of a homotopy-invariant algebraic intersection number (the so-called “star-pairing”), the notion of hidden intersections at infinity and asymptotic positivity of intersections, and a punctured generalization of the adjunction formula.